Conservation of energy for toy gun

I've been reading about a situation on the conservation of energy:
A spring-loaded toy gun is used to shoot a ball of mass m straight up in the air. View Figure The spring has spring constant k. If the spring is compressed a distance x_0 from its equilibrium position and then released, the ball reaches a maximum height h_max (measured from the equilibrium position of the spring). There is no air resistance, and the ball never touches the inside of the gun. Assume x_o is greater than h

Now my question to this is that:
Is mechanical energy conserved because no nonconservative forces perform work on the ball and do nonconservative forces act in this situation after the ball is released at all? and do the forces of gravity and the spring have potential energies associated with them?

As you can tell I don't really have a great grasp on these concepts, so would anyone like to enlighten me please???? thanks so much!!

Staff: Mentor

WY said:

Now my question to this is that:
Is mechanical energy conserved because no nonconservative forces perform work on the ball and do nonconservative forces act in this situation after the ball is released at all?

No nonconservative forces act on the ball at any time (in this problem). That's why they specify "There is no air resistance, and the ball never touches the inside of the gun."; those would be nonconservative forces.

and do the forces of gravity and the spring have potential energies associated with them?

Absolutely. Gravitational potential energy (near the earth's surface) is given by [itex]mgh[/itex], where "h" is height measured from some arbitrary reference point. Spring potential energy is given by [itex]1/2 k x^2[/itex], where x is the displacement from the unstretched position.

so, h,max=(1/2)kx^2/gm
look closely at this equation and it will make sense to you!
the stiffer the spring the larger the "k" the more energy the spring has when it is compressed, and the higher the ball will go for a given x the distance the spring is compressed! A springs potential energy stored is proportional [1/2 k] to its distance compressed squared.
and inversely proportional to both the earth's gravitational constant and the mass of the object shot up!

"A springs potential energy stored is proportional [1/2 k] to its distance compressed squared. and inversely proportional to both the earth's gravitational constant and the mass of the object shot up! this should make some sense, right?"

Actually no.

What does "the earth's gravitational constant" or "the mass of the object shot up" have to do with a spring's potential energy?